The present invention, in some embodiments thereof, relates to methods and systems for managing networks and, more specifically, but not exclusively, to methods and systems for approximating solutions for optimized management of a liquid distribution networks such as a pressurized water distribution network.
Water distribution networks are complex entities, composed of many types of components, such as pipes, valves, pumps, and tanks, making their efficient management a significant challenge. The growing worldwide demand for water as well as increasing urbanization renders a growing complexity of water distribution networks. Managing large complex water distribution networks requires coping with water distribution network matters such as: water stagnation management, demand prediction, supply to end consumers according to predefined requirements (pressure, cost, quantity and quality), reduction of Non-Revenue Water (NRW) etc.
One of the most significant matters in water distribution network management, in terms of cost and environmental impact, is reducing Non-Revenue Water (NRW). NRW is water that is input into the network but is not paid for or generating revenue. NRW includes water lost due to leaks, bursts and theft as well as water retention in tanks. According to the Environmental Protection Agency, much of the 880,000 miles of water pipes in the United States has been in service for decades—some for over 100 years—and can be significant source of water loss. The World Bank estimates that worldwide costs from leaks total 14 billion United States Dollars annually.
Reducing the water pressure in the network is a well-known recommended practice for reducing NRW. The higher the pressure in the network, the larger the amount of water lost in leaks and bursts. Consequently, lowering the pressure in the water network has the potential to significantly reduce water loss. However, there are additional, often conflicting, goals associated with pressure management: 1) Water supply pressure—A high enough pressure level must be maintained to ensure that water reaches all consumers at the required flow rates. 2) Water turnover—Water pressure has a direct influence on water turnover—the length of time during which water remains in tanks. A higher water pressure causes water to remain in tanks for longer periods, which might lead to quality issues due to water stagnation or the decay of disinfectants like chlorine. In addition to these conflicting goals, there are several other factors that contribute to the challenge of water pressure management: i) The pressure in the system is highly dependent on the demand for water. However, such demand varies both during the day and across seasons. ii) In highly connected networks, changing one valve or pump or tank setting in order to adjust pressure in one part of the system may actually have a detrimental impact on the pressure in another part of the system.
Managing a pressurized water distribution system is a challenging task. Solution approximation is often sought by engineers based on intimate familiarity and experience with a particular water distribution network. This type of intuition-based strategy is error prone and often results in sub optimal management. Often solutions achieved by formal computerized methods surplus intuition-based strategies in performance. Current computerized methods for approximating a solution aim to find feasible solutions utilizing strategies such as: Linear programming, non-linear programming, evolution based heuristics and mixed integer nonlinear programming.
Liner programming is often applied to complex networks, such as a pressurized water distribution network, in its mixed integer specimen: mixed integer linear programming (MILP). However, pressurized water distribution networks are not linear in nature. Pressurized water distribution network aspects such as head loss formulas and pumps discharge equations are not linear (Methods, H., Walski, T., Chase, D., Savic, D., Grayman, W., Beckwith, S., Koelle, E., 2003, “Advanced water distribution modeling and management”, Bentley Institute Press). In order to apply MILP to pressurized water distribution network, the problem is linearized. The linearization is performed in a manner assuring a high quality solution can be found within a reasonable running time. Linearization is performed, for example by the piece-wise linearization technique (Bertsimas, D., Tsitsiklis, J. N, 1997, “Introduction to linear optimization”, Athena Scientific Belmont). Piece-wise linearization covers by small enough linear pieces a non-linear curve when a solver chooses only one of them as a solution. A drawback of this technique is long running times for real scale problems (Eck, B., Mevissen M., “Valve Placement in Water Networks: Mixed-Integer Non-Linear Optimization with Quadratic Pipe Friction”, IBM Research Report). Another linearization technique is based on the first order Taylor series approximation (Sherali D., Smith, E., 1997, “A Global Optimization Approach to a Water Distribution Network Design Problem, Journal of Global Optimization”, 11(2), 107-132). Although first order Taylor series approximation technique by itself is faster than piece-wise linearization technique, it is unlikely to be applied to large scale problems such as real cities pressurized water distribution networks due to related long running time and inferior solution quality. Another linear programming method is a sequential linear programming (Sterling, M., Bargiela, A., 1984, “Leakage Reduction by Optimized Control of Valves in Water Networks”, Transactions of the Institute of Measurement and Control, 6(6), 293-298). Sequential linear programming starts with an initial guess of the solution (decision variables). Then, it is iteratively solves the problem, updating the solution at each iteration and using the current iteration solution for developing first order Taylor series for the next one. In each iteration, a linear program which represents a whole network must be solved. As multiple iterations are required, overall running times are expected to be unfeasible for large scale problems such as pressurized water distribution networks. Moreover, the solution quality of sequential linear programming is compromised by approximation and integer variables handling.